Adaptive Control of Linear Periodic Systems Using Multiple Models

Even though the raison d'etre of adaptive control is to cope with time-varying environments, for the sake of mathematical tractability researchers have traditionally confined their attention to time-invariant (or slowly time-varying) systems. The limitations of classical adaptive control, however, become evident when the controllers are called upon to respond to rapidly varying environments. In recent years numerous nontechnical applications such as medical emergencies, trading on the stock market, conflict management using counter terrorism measures, and technical applications such as aircraft and automobile control, energy management, and manufacturing are arising which call for fast and accurate control in such environments. One of the main difficulties while dealing with time-varying environment is in characterizing them appropriately, so that the problems posed lend themselves to mathematical analysis. In this paper we attempt to combine multiple fixed and adaptive models in a hierarchical approach to achieve fast and accurate response, when the parameters of a plant vary periodically in an unknown fashion. Theoretical analyses followed by simulation studies of increasingly complex static and dynamical systems with single and multiple parameters are presented. Many of the theoretical questions addressed in this paper in the specific context of systems with periodic parameters are found to be relevant for adaptation in more general time-varying environments.